Integrality conjecture for TQFT invariants and cut number

Let $M$ be an oriented connected closed three-manifold, possibly equipped with a banded trivalent colored graph, and let $p$ be a prime. Define

$$I_p(M)=\mathcal{D}_p\langle\langle M\rangle\rangle_p\in {{\mathbb{D}}}_p.$$

For a connected three-manifold $M$, let $c(M)$ denote its cut number: the maximal number of oriented surfaces that can be placed in $M$ while keeping the complement connected. Integrality conjecture. The element $I_p(M)$ is divisible by

$$\mathcal{D}_p^{c(M)}.$$

The conjecture predicts that the TQFT invariant has increasing divisibility with the cut number, relating quantum invariants to the topology of the fundamental group through the cut number. The source does not state a resolution.